Volume 9 (2002), Number 2, 271–286
ON AN OPTIMAL DECOMPOSITION IN ZYGMUND SPACES
ALBERTO FIORENZA AND MIROSLAV KRBEC
Abstract. An optimal decomposition formula for the norm in the Orlicz space L(logL)α is given. New proofs of some results involving L(logL)α spaces are given and the decomposition is applied to apriori estimates for elliptic partial differential equations with the right-hand side in Zygmund classes.
2000 Mathematics Subject Classification: 46E35, 46E39, 35J05.
Key words and phrases: Orlicz spaces, Zygmund classes, extrapolation, duality.
1. Introduction
A lot of attention has always been paid to various expressions for the norm in Orlicz spaces (Luxemburg norm, “dual” norm, Amemyia norm, see, e.g., [24], [28], [25]); indeed, the definition of a norm is one of very peculiar features of these spaces, which sometimes makes things rather difficult and – by the rule of thumb – frequently prevents one from straightforward usage of the Lp-space
techniques and approach.
In general, the idea of decomposition has turned out to be highly fruitful in many areas of analysis. It is often helpful to consider functions decomposed into suitable elementary pieces which are easier to handle. In connection with extrapolation procedures there is even a chance to consider not only these small pieces in a given space, but also to treat them as elements of spaces “in prox-imity” of the considered space. We shall see that it is natural to arrive at L(logL)α via the scale ofLp-spaces withp > 1, breaking functions into suitable
little pieces belonging to “better” Lp-spaces.
This has been done at the abstract extrapolation theory level in Jawerth and Milman [22], see also Milman [26]. The identification of the result of the Σ-method applied to Lp-spaces is done via including the K-functional into the
considerations and then employing the fact that Zygmund spaces can also be interpreted as special Lorentz–Zygmund spaces (see [1, Chapter 4]).
A functional analysis argument leads to the same expression in Edmunds and Triebel [7] for the range 1 < p < ∞. Here, the key estimate for extrapolation characterization ofLp(logL)−α, 1< p <∞andα >0, is the estimatec
1kfkp1 ≤
kfkp1,p ≤c2kfkp2 with suitable p1 < p2 < p, relating the Lorentz and Lebesgue norms. Then a duality argument is used for spaces Lp(logL)α,α >0. The case
p= 1 in [8] requires an extra effort since Lexpt1/α is not the predual ofL(logL)α
(cf. also Remark 3.7). In this connection let us also recall Fiorenza [9], where it has been shown in another situation that in some cases the norm in a dual space can be expressed in terms of a suitable decomposition.
The plan of the paper is the following: In Section 2 we introduce the no-tation and recall some relevant result framing our considerations. Section 3 contains the main results and their proofs. In the concluding Section 4 we give some examples showing how the decomposition norm can be employed to get straightforward proofs of some results involving LlogL-spaces and some new apriori estimates for elliptic partial differential equations.
2. Preliminaries
In the sequel we assume thatΩ is an open bounded subset ofRN of Lebesgue measure|Ω|= 1. If not otherwise specified, all the norms and all the spaces are tacitly assumed to concern Lebesgue measurable functions defined a.e. in Ω. All positive constants whose exact value is not important for our purposes are denoted by c, occasionally with additional subscripts within the same formula or the same proof.
Let either α ∈ R and 1 < r < ∞ or α ≥ 0 and r = 1. The Orlicz space
Lr(logL)α is defined as the Banach space of all measurable functions f on Ω such that the following (Luxemburg) norm
kfkLr(logL)α = inf
(
λ >0 : Z
Ω
µ|f| λ
¶r logα
µ e+
µ|f| λ
¶r¶
dx <1 )
is finite. If α = 0 then the space Lr(logL)α reduces to the Lebesgue space Lr
whose norm is denoted simply by k · kr. If r = 1 and α > 0, we do speak
about the Zygmund classes. We shall not discuss the case r =∞, α <0; this leads to exponential Orlicz spacesLexpt1/α
(and gives justification to alternative notation L∞(logL)α for them). Sometimes another notation is employed: for
β >0, the space EXPβ (or simply EXP ifβ = 1) is defined as the Banach space
of all measurable functions f onΩ such that
kfkEXPβ = inf
(
λ >0 : Z
Ω
e(|f|/λ)βdx <2 )
is finite. Later we shall make use of the well-known extrapolation characteriza-tion of EXPβ as of the space consisting of all f such that
sup
k∈N kfkk
k1/β <∞ (2.1)
(see, e.g., [14, Exercise 17, p. 279]). Observe that the sup on the left hand side is an equivalent norm in EXPβ.
We use the notation λj = (2j)′ = (1−2−j)−1, j ∈ N. For later use let us
J ∈N, then the functional
kgkr,α = inf
|g|=P∞j=Jgj
µ ∞ X
j=J
2jαkgjkrrλj
¶1/r
(2.2)
defines a norm in Lr(logL)α equivalent to the Luxemburg norm. Clearly, any
sequence (gj) such that for somej the functiongj is not inLrλj makes no
contri-bution to the computation of the infimum, therefore without loss of generality we may assume thatgj ∈Lrλj for allj ≥J. Plainly, we can assume thatJ = 1.
Also, we shall restrict ourself to representations of |g| as sums of non-negative functions gj and we shall consider only non-negative functions in more general
representations of type |g|1/q =P
|gj|1/q (q≥1) later; it is easy to see that the
infimum remains to be the same (provided the powers have sense). We call such decompositions admissibleand in the sequel all the infima over decompositions of |g|will be taken over admissible decompositions without an explicit recalling this assumption. Note that some more delicate considerations about similar decompositions into sign-changing functions can be found in [9].
The non-increasing rearrangement of a non-negative measurable function f on Ω is defined as f∗(t) = inf{λ > 0; m(f, λ) ≤ t}, t > 0, where m(f, λ) =
|{x ∈ Ω; f(x) > λ}|, or, alternatively, in terms of the function f, as f∗(t) =
sup|E|=tess inf
x∈E f(x),t∈(0,1] (see [12]). These definitions yield functions, which
are equal up to a countable subset of (0,1), thus the difference between them can be neglected as far as their integration is concerned. It can be easily proved that f∗ is non-negative and non-increasing on (0,1) and vanishing for t ≥ 1.
Moreover, Orlicz spaces are rearrangement-invariant, that is, the norm of a function in an Orlicz space is preserved when considering its non-increasing rearrangement. For more details about non-increasing rearrangements we refer to [1], [19], [33], [34].
We refer to [23], [28], [16], and [24] for the theory of Orlicz spaces and integral operators acting on them.
Remark 2.1. Let us recall, for completeness, various known formulas for norms of a function f ∈ L(logL), all of them equivalent to the Luxemburg norm of f; for simplicity, we restrict ourselves here to the case Ω = [0,1]. The list follows:
inf (
λ >0 :
1
Z
0
(|f(t)|/λ) log(A+ (|f(t)|/λ))dt < B )
for any A≥1, B >0;
sup ( 1
Z
0
f(t)g(t)dt: kgkEXP ≤1
)
(the Orlicz norm, or the dual norm);
1
Z
0
µ1
t
t
Z
0
f∗ ¶
1
Z
0
f∗(t) log(1/t)dt;
1
Z
0
f(t) log µ
e+ f(t)R1
0 f
¶ dt;
inf (
1 λ +
1
Z
0
λ|f|log(e+λ|f|)dx: λ >0 )
(the Amemyia norm).
The first expression is clear from the well-known imbedding theorems in Orlicz spaces and the second can be found, e.g., in [24]. The norms in the third and fourth line in terms of f∗ appear for instance in [1, Chapter 4], and their
equivalence follows from Herz’s theorem (see, e.g., [1, Chapter 3, Theorem 3.8]). The last two expressions for the norm can be found, for instance, in [3] and [25].
Remark 2.2. The norms R1
0 f∗(t) log(1/t)dt above, and/or, more generally,
R1
0 f∗(t)[log(1/t)]αdtinL(logL)αgive information about the right compensation
for the growth rate of theLp-norm of a function inL(logL)α asp→1. Namely,
we have
1
Z
0
f∗(t)hlog(1/t)iαdt ≤ kfkp
à 1 Z
0
³
log(1/t)´αp ′
dt !1/p′
=kfkp
à ∞ Z
0
ξαp′e−ξdξ !1/p′
=kfkp[Γ(αp′+ 1)]1/p
′
=kfkp(αp′)1/p
′
[Γ(αp′)]1/p′,
where Γ is the gamma-function. According to Stirling’s formula we have
[Γ(αp′)]1/p ∼h√2π(αp′)αp′−1/2e−αp′i1/p ′
∼(p′)α−1/(2p′) ∼(p′)α as p′ → ∞. Altogether
kfkL(logL)α ≤c(p′)αkfkp as p→1 (2.3)
with some c independent ofp and f.
The following lemma is a simple observation about the relation between Lux-emburg norms in different Orlicz spaces. The functionΦappearing in the state-ment can be any Young function (i.e., any real function on [0,∞), continuous, increasing, convex, and such that lim
t→0Φ(t)/t = limt→∞t/Φ(t) = 0).
Lemma 2.3. For any p≥1 and for any function f ∈LΦ(tp)
we have
kfkΦ(tp) =kfpk1Φ/p(t).
3. A decomposition Norm
Let Mdenote the set of all Lebesgue measurable functions onΩ.
Theorem 3.1. For any q ≥1 and α≥0, the functional
g 7→ kgk1,α,q = inf
|g|=
µ P∞
j=1g 1/q j ¶q ∞ X j=1
2jαkg jkλj
has the following properties:
(i) kgk1,α,q ≥0, g ∈ M;
(ii) kλgk1,α,q =|λ|kgk1,α,q, g ∈ M;
(iii) g ∈ M, kgk1,α,q = 0 ⇔g = 0 a.e. in Ω;
(iv) kg+hk1,α,q ≤ kgk1,α,q+khk1,α,q, g, h∈ M.
Proof. The proof of (i) is immediate, and (ii) is easy. As to (iii), we prove that if g ∈ M, kgk1,α,q = 0, then g = 0 a.e. in Ω.
Let us consider first the easy case q = 1. Given ε >0, there exists (gj) such
that
kgk1 =
° ° ° ∞ X j=1 gj ° ° °1 ≤
∞
X
j=1
2jαkgjkλj < ε
and therefore we get the assertion. If q > 1, we make use of the following argument (in which, of course, we may assume g ≥ 0) which we develop now only for α= 1, q= 2; the remaining cases can be treated similarly.
Fix 0 < H < 1 and a decomposition g = ³P∞
j=1g 1/2
j
´2
. We claim that ¯
¯
¯{g > 20H} ¯ ¯ ¯ ≤ ¯ ¯ ¯ S∞
j=1{gj > (2/3)jH}
¯ ¯
¯. Indeed, if this is not true, then there exists a set E ⊆ Ω of positive measure such that g(x) > 20H for a.e. x ∈ E, gj(x)≤(2/3)jH for a.e. x∈E and for every j ∈N. Therefore
g = µ ∞
X
j=1
gj1/2
¶2 ≤ · ∞ X j=1 ³
(2/3)jH´1/2 ¸2
<20H for a.e. x∈E,
which is absurd. Our claim is proved and therefore we have also
¯ ¯
¯{g >20H} ¯ ¯ ¯≤ ∞ X j=1 ¯ ¯
¯{gj >(2/3)
jH}¯ ¯ ¯.
Now we estimate |{gj >(2/3)jH}| for a particular choice of (gj). Let us fix
0< ε <1, and denote again by (gj) a sequence such that
g = µ ∞
X
j=1
g1j/2
¶2 and
∞
X
j=1
2jkgjkλj < ε.
We have 2jkg
jkλj < ε, j ∈N, and therefore
2jλj¯¯
¯{gj >(2/3)
jH
}¯¯ ¯ ³
(2/3)jH´λj ≤2jλj Z
{gj>(2/3)jH}
Our estimate follows because
¯ ¯
¯{gj >(2/3)
jH
}¯¯ ¯≤ε2
−jλjµ 3j
2jH
¶λj
≤ ε(3/4)
j
H2 , j ∈N.
Hence
¯ ¯
¯{g >20H} ¯ ¯ ¯≤ ∞ X j=1 ¯ ¯
¯{gj >(2/3)
jH }¯¯ ¯≤ ε H2 ∞ X j=1 µ3 4 ¶j
≤ H3ε2.
This inequality implies |{g >20H}|= 0 for all 0< H <1. Therefore g = 0 for a.e. x∈Ω and (iii) is proved.
Finally, we prove the triangle inequality (iv). First, we have
inf
|g+h|≤³P∞
j=1z 1/q j ´q ∞ X j=1
2jαkzjkλj ≤ inf
|g|=³P∞
j=1g 1/q j
´q
|h|=
³ P∞
j=1h 1/q j ´q ∞ X j=1
2jαkgj+hjkλj.
This is easy to see: if (gj), (hj) are such that
|g|= µ ∞
X
j=1
gj1/q ¶q
and |h|= µ ∞
X
j=1
h1j/q ¶q
,
then for zj =gj+hj,j ∈N, we have
|g+h| ≤ |g|+|h| ≤ µ ∞
X
j=1
zj1/q ¶q
,
where we used Minkowski’s inequality with an exponent between 0 and 1 (see, e.g., [19, n. 25, (2.11.5), p. 31]). Now we have
kg+hk1,α,q = inf
|g+h|=³P∞
j=1z 1/q j ´q ∞ X j=1
2jαkzjkλj = inf
|g+h|≤³P∞
j=1z 1/q j ´q ∞ X j=1
2jαkzjkλj
≤ inf
|g|=³P∞
j=1g 1/q j
´q
|h|=
³ P∞
j=1h 1/q j ´q ∞ X j=1
2jαkgj +hjkλj
≤ inf
|g|=³P∞
j=1g 1/q j
´q
|h|=³P∞
j=1h 1/q j ´q ∞ X j=1
2jαkgjkλj +
∞
X
j=1
2jαkhjkλj
= inf
|g|=³P∞
j=1g 1/q j ´q ∞ X j=1
2jαkgjkλj + inf
|h|=³P∞
j=1h 1/q j ´q ∞ X j=1
2jαkhjkλj
=kgk1,α,q+khk1,α,q
Next step is to prove that the norm defined in Theorem 3.1 is equivalent to the Luxemburg norm in L(logL)α. We shall make use of
Lemma 3.2. Let Φbe any Young function. If N(f)is a norm in X ={f ∈ L1 : N(f)<∞} and if
f ∈L1 7→hN(|f|p)i1/p
is a norm in LΦ(tp)
equivalent to the Luxemburg norm, then X =LΦ(t) and
f ∈L1 7→ N(f)
is a norm in LΦ(t) equivalent to the Luxemburg norm.
Proof. We know that there existc1, c2 >0 such that
c1kfkΦ(tp) ≤[N(|f|p)]1/p≤c2kfkΦ(tp), f ∈LΦ(t p)
.
Therefore c1k|f|1/pkΦ(tp) ≤ [N(f)]1/p ≤ c2k|f|1/pkΦ(tp), f ∈ LΦ(t). In view of
Lemma 2.3, c1kfk1Φ/p(t) ≤ [N(f)]1/p ≤ c2kfk1Φ/p(t), f ∈ LΦ(t). Hence c
p
1kfkΦ(t) ≤
N(f)≤cp2kfkΦ(t),f ∈LΦ(t).
Proposition 3.3. Let q > 1. Then k · k1,α,q is equivalent to the Luxemburg norm of L(logL)α.
Proof. We have
kgqk11/q,α,q = inf
|g|q=³P∞
j=1g 1/q j
´q µ ∞
X
j=1
2jαkgjkλj
¶1/q
= inf
|g|=P∞
j=1γj µ ∞
X
j=1
2jαkγjqkλj
¶1/q
= inf
|g|=P∞
j=1γj µ ∞
X
j=1
2jαkγjkqqλj
¶1/q .
By (2.2) the last quantity is equivalent to the Luxemburg norm in Lq(logL)α.
Therefore, by virtue of Lemma 3.2, we get the assertion.
Our next step will be to prove that Proposition 3.3 remains true for q = 1. At the same time we find an optimaldecomposition of g into a sum; this is the core of the paper. We observe that one inequality of the equivalence of kgk1,α,1
to the Luxemburg norm can be proved as a corollary of Proposition 3.3. We shall use, however, another argument, independent of [7], and we prefer to prove both inequalities in the desired equivalence in a direct way, giving us additional information about how to decompose a function in order to get an expression equivalent to the norm in L(logL)α. Let us recall the well-known fact that
f ∈EXPβ if and only if supk∈Nk−1/βkfkk <∞ and that the latter expression
is an equivalent norm in EXPβ. (See, e.g., [14, Chapter VI, Exercise 17].)
Moreover, this statement can be improved (see [6]), namely, f ∈ EXPβ if and
only if supk∈Nk−1/βkf χIkkk<∞, whereχIk is the characteristic function of the
interval (ek, ek−1), k = 1,2, . . . (and one gets an equivalent norm in this way,
Theorem 3.4. Let α >0. Then
kgk1,α,1 = inf |g|=P∞j=1gj
∞
X
j=1
2jαkgjkλj (3.1)
and the Luxemburg norm are equivalent in L(logL)α.
Proof. We may assume that g ≥ 0 and without loss of generality, we consider the case Ω = (0,1) ⊂RN1. Moreover, in order to show the first inequality we may consider the “dual norm” in L(logL)α instead of the Luxemburg norm:
kgkL(logL)α = sup
( 1 Z
0
f(t)g(t)dt: kfkEXP1/α ≤1
) .
Fix any decomposition g = P∞
j=1gj. If g ∈ L(logL)α, then for any ε > 0 let
fε∈EXP1/α, kfεkEXP1/α ≤1,fε≥0, be such that
kgkL(logL)α −ε≤
1
Z
0
fεg dx.
Ifg /∈L(logL)α, then one can put any positive number instead ofkgk
L(logL)α−ε.
Then
kgkL(logL)α −ε=
∞
X
j=1 1
Z
0
fεgjdx=
∞
X
j=1 1
Z
0
fε
2jα2 jαg
jdx
≤
∞
X
j=1
° ° ° °
fε
2jα
° ° ° °
2j2 jα
kgjkλj ≤ kfεkEXP1/α
∞
X
j=1
2jαkgjkλj ≤
∞
X
j=1
2jαkgjkλj,
where the last but one inequality follows from the standard extrapolation char-acterization of EXP1/α (see (2.1)). This yields
kgkL(logL)α ≤
∞
X
j=1
2jαkgjkλj.
Since the decomposition of g asg = P∞
j=1gj can be arbitrary, we also have
kgkL(logL)α ≤ inf
|g|=P∞
j=1gj
∞
X
j=1
2jαkgjkλj.
On the other hand, if g ∈ LlogL, then we have the following estimate, in which k′ denotes the H¨older conjugate exponent of k. To avoid unnecessary
technicalities and clumsy formulas, for k = 2 read (k − 1)′ in the following
estimates as any number greater than 4′ = 4/3. We have
kgkL(logL)α ≥c
1
Z
0
≥c
∞
X
k=2
e−k+1 Z
e−k
g∗(t)[log(1/t)]αdt≥c
∞
X
k=2
kαg∗(e−k+1)e−k
≥c ∞ X k=2 kα µ e
−k+2
Z
e−k+1
[g∗(t)](k−1)′dt ¶ 1
(k−1)′
e−khe−k+1(e−1)i− 1 (k−1)′
≥c ∞ X k=2 kα µ e
−k+2 Z
e−k+1
[g∗(t)](k−1)′dt ¶(k−11)′
=c
∞
X
j=1 2j+1−1
X
k=2j
kα µ e
−k+2 Z
e−k+1
[g∗(t)](k−1)′dt ¶ 1
(k−1)′
≥c
∞
X
j=1 2j+1−1
X
k=2j
2jα µ e
−k+2
Z
e−k+1
[g∗(t)](k−1)′dt ¶ 1
(k−1)′
≥c
∞
X
j=1
2jα
2j+1−1 X
k=2j
µ e
−k+2 Z
e−k+1
[g∗(t)](k−1)′dt ¶(k−11)′
.
Now, by H¨older inequality,
à e
−k+2
Z
e−k+1
[g∗(t)](k−1)′dt !(k−11)′
≥ Ã e
−k+2 Z
e−k+1
[g∗(t)](2j+1)′dt
! 1
(2j+1)′
h
e−k+2−e−k+1i[(2
j+1)′
/(k−1)′ ]−1
.
Elementary estimates show that the second term on the right-hand side is equiv-alent to a positive constant as k → ∞. Hence the norm in L(k−1)′
can be es-timated by the norm in L(2j+1)′
from below for j = 2j, . . . ,2j+1 −1 and the
constant in the estimate remains the same for all k ∈ N. Now it suffices to apply the triangle inequality to conclude that
kgkL(logL)α ≥c
∞
X
j=1
2jα µ e
−2j+2
Z
e−2j+1+2
[g∗(t)](2j+1)′dt ¶ 1
(2j+1)′
≥c inf
|g|=P∞
j=1gj
∞
X
j=1
2jαkgjkλj.
Corollary 3.5. Let α >0. Then
kfkL(logL)≈ ∞
X
j=1
2jαkf∗k(2j)′,Jj, (3.2)
where (2j)′ = 2j/(2j −1), J
j = (e−2 j+1+2
, e−2j+2
), j = 1, . . ..
Remark 3.6. After proving Theorem 3.4 it is clear that every member of the last chain of inequalities of the proof is equivalent to the Luxemburg norm in LlogL. The decomposition in (3.2) is one of them and since it can be considered as a particular argument of the infimum in (3.1), we have actually found one of the “best”, or “optimal” decompositions realizing the infimum in (3.1) up to an equivalence. Note that we do not know whether there exist decompositions which realize exactly the infimum.
Remark 3.7. Expression (3.2) permits, in particular, an interpretation of the norm in LlogL as a norm of a special sequence in ℓ1.
Let us recall the following notation for the spaces of sequences: c0is the space
of all numerical sequences converging to zero, equipped with the sup norm, ℓ1
is the space of all numerical absolutely convergent sequences (and the sum of the series is the norm),ℓ∞is the space of all numerical bounded sequences with
the sup norm. The links between spaces of functions and spaces of sequences could be illustrated by the following scheme, also summarizing the other known results. We shall denote by exp the closure of L∞ in EXP. Then we have
g ∈exp f ∈LlogL h∈EXP
m m m
kgkk,Ik
k ∈c0 2
jkf∗k
(2j)′,Ij ∈ℓ1 k hkk,Ik
k ∈ℓ∞
These three equivalencies express some characterizations of the spaces in ques-tion. For the first one in “global” terms of k−1kgk
k, see [4], [15], [13], [29]. Let
us note that this follows directly in terms of k−1kgk
k and k−1kgkk,Ik from the
fact that supk−1kfk
k and supk−1kfkk,Ik respectively is an equivalent norm in
EXP. Indeed, if g ∈ exp, then for any ε > 0 there exists h ∈ L∞ such that
supk−1kg −hk
k,Ik ≤ ε. Since k−1khkk,Ik → 0 as k → ∞, we get k−1kgkk,Ik ≤
k−1kg−hk
k,Ik+k−1khkk,Ik < εfor largek. The second equivalence is contained
in our Theorem 3.4, and the third one is due to Edmunds and Krbec ([6]). Observe that the spaces in the first and the second row are the preduals of the spaces occurring in the second and the third row, respectively.
4. Applications
Theorem 4.1. If A is a sublinear operator bounded in Lp, 1< p≤p
0 <∞
such that
kAfkp ≤c1p1/βkfkp, f ∈Lp, p >1,
for some β >0andc1 independent of pandf, then Af ∈EXPβ for allf ∈L∞.
After our Theorem 3.4 we can prove a “dual” result, namely, we derive esti-mates of operators in logarithmic spaces starting from analogous estiesti-mates in Lebesgue spaces. A central role is played by the order of infinity of the constant in the Lp estimates.
Theorem 4.2. If H is a subadditive operator acting on L1 and such that
kHgkr(p)≤
c1
(p−1)βkgkp, g ∈L
p, 1< p
≤p0 <∞,
for some constants c1 > 0, β ≥ 0, 1 < p0 < ∞, independent of p, g and for
some r : [1, p0] → [1,∞) such that p ≤ r(1)p ≤ r(p), for every p ∈ [1, p0],
independent of g, then there is c2 >0 such that for any α≥β,
kHfkLr(1)(logL)r(1)(α−β) ≤c2kfkL(logL)α, f ∈L∞, f ≥0,
the inequality is true with c2 =c1 if the norms in the logarithmic spaces
consid-ered are given by (2.2) and (3.1).
Proof. Given f ∈L∞, f ≥0, fix a decomposition f =P∞
j=1fj, fj ≥ 0, j ∈ N.
For any ε > 0 there exists ν ∈ N such that we have, by using the inequality r(1)≤r(1)p0 ≤r(p0) and the assumption fj ≥0,j ∈N,
kHfkLr(1)(logL)r(1)(α−β) ≤ ° ° ° ° ° H µ ν X j=1 fj !° ° ° °
°Lr(1)(logL)r(1)(α−β) +c ° ° ° ° ° H µ ∞ X
j=ν+1
fj
!° ° ° ° °r(p
0) ≤ ° ° ° ° ° H µ ν X j=1 fj !° ° ° °
°Lr(1)(logL)r(1)(α−β)
+ cc1
(p0−1)β
° ° ° ° ° ∞ X
j=ν+1
fj ° ° ° ° ° p0 ≤ ° ° ° ° ° H µ ν X j=1 fj !° ° ° ° °
Lr(1)(logL)r(1)(α−β) +ε.
Therefore
kHfkLr(1)(logL)r(1)(α−β) ≤ sup
m∈N ° ° ° ° ° H µ m X j=1 fj !° ° ° ° °
Lr(1)(logL)r(1)(α−β)
≤ sup
m∈N ° ° ° ° m X j=1
Hfj
° ° ° °
°Lr(1)(logL)r(1)(α−β)
≤ µ ∞
X
j=1
2r(1)(α−β)j ° ° ° ° °
Hfj
° ° ° ° ° r(1)
r(1)λj
!1/r(1)
≤
∞
X
j=1
2(α−β)jkHf
jkr(λj)≤c1 ∞
X
j=1
2(α−β)j
(2−jλ j)βk
≤c1 ∞
X
j=1
2αjkfjkλj,
where the passage from the third line to the fourth one is justified by (2.2). Observe, however, that one can drop the fourth line completely – the corre-sponding argument for passing from the third line to the fifth one is estimate (2.3). Taking the infimum over all admissible decompositions of f, we get the assertion.
Let us recall the (local) maximal operator (see, e.g., [10]). For any measurable function f on Ω, let
M f(x) = sup
x∈Q
1
|Q| Z
Q
|f(y)|dy, x∈Ω,
where the supremum is taken over all cubes Q in Ω containingx and with the sides parallel to the coordinate axes. The following Hardy–Littlewood–Wiener maximal theorem is well known:
kM fkp ≤c(p)kfkp, f ∈Lp, 1< p ≤ ∞,
where c(p) = O((p−1)−1) as p → 1 (See, e.g., [32, Chapter 1], for the
quan-titative behaviour of the norms resulting from the interpolation of weak type operators.)
Applying Theorem 4.2 to the maximal operator in the case r(p) =p, β = 1, we get the following well-known result (see [31], [32, p. 23]), as a consequence of the Hardy–Littlewood–Wiener maximal theorem and of the a.e. monotone convergence properties of the maximal operator:
Corollary 4.3. Let α ≥0. Then there is c >0 such that
kM gkL(logL)α ≤ckgkL(logL)α+1 for all g in L(logL)α+1.
We observe that it is known that Corollary 4.3 follows from an extrapolation procedure (see, e.g., [33]), but we point out here that the proof of Theorem 4.2 follows immediately from the definition of the norm in LlogL in terms of de-composition.
A result equivalent to the previous one (see, e.g., [1]), that can be proved independent of the concept of a non-increasing rearrangement (see also [19]), is the limiting case of the Hardy inequality as p → 1. We shall get it as a consequence of Theorem 4.1 when applied to the Hardy operator
f 7→T f = 1 x
x
Z
0
f(t)dt, x∈(0,1)⊂R1,
whose boundedness in Lp-spaces is given by the classical Hardy inequality
kT fkp ≤
p
Corollary 4.4. For any f in L(logL)α+1(0,1), α≥0, we have
kT fkL(logL)α(0,1) ≤ckfkL(logL)α+1(0,1)
with c >0 independent off.
Let us now give another application of our extrapolation procedure in the the-ory of partial differential equations, and let us consider as a model the Dirichlet problem on a bounded open set Ω ⊂RNn, n≥3, with C1,1-boundary
divA(x)∇u=f in Ω,
u= 0 on ∂Ω, (4.1)
where f ∈ Lp(Ω), 1 < p < n, and A = (a
ij)i,j=1...n is such that aij ∈
VMO∩L∞(RNn),a
ij =aji and there exist 0< λ0 ≤Λ0 <∞ such that
λ0|ξ|2 ≤
X
aijξiξj ≤Λ0|ξ|2
for all ξ ∈ RNn and for a.e. x ∈ Ω. For the sake of simplicity, we shall also assume that |Ω|= 1.
It is well known (see [2]) that there exists one and only one (weak) solution u to the b.v.p. (4.1) in the Sobolev space W01,np/(n−p)(Ω). Moreover, this is not
true when p= 1 (see [11], [17]).
Iff ∈L(logL), the existence of a solution uinW01,n/(n−1) can be proved (see
[30]), and actually a better estimate of the norm of Du has been proved ([27]) by using duality techniques:
k|Du|kLn/(n−1)
(logL)−1+[nα/(n−1)] ≤KkfkL(logL)α (4.2) for all 0 < α ≤ 1. As an application of our Theorem 4.2, we prove estimate (4.2) for any α ≥ (n−1)/n, extending thus the validity of (4.2) for all α >0. We shall use the following result, due to Di Fazio ([5], see also [20], [21]):
Theorem 4.5. Under the above assumptions on A, if |F| ∈Lp, 1< p <∞, then the Dirichlet problem
divA(x)∇u= divF in Ω,
u= 0 on ∂Ω,
has a unique solution and, moreover, there exists a constant c such that
k|∇u|kp ≤ck|F|kp. (4.3)
We note that the constantcin (4.3) can be chosen independent ofpifpstays away from 1 and ∞.
We can prove now
Lemma 4.6. Let 1< p < n, f ∈Lp, and let u be a solution of (4.1). Then
k|∇u|knp/(n−p) ≤
c
Proof. Let us consider a solution F to the equation divF = f, which can be expressed explicitly in terms of the vector Riesz potential
F(x) = 1 nωn
Z
Ω
x−y
|x−y|nf(y)dy,
where ωn is the measure of the unit ball in RNn (see [18]). By using the
well-known estimates which come out from the Hardy–Littlewood–Wiener maximal theorem and the Sobolev inequality for Riesz potentials (see [34]) we get
k|F|knp/(n−p) ≤
c1
(p−1)(n−p)/nkfkp ≤
c2
(p−1)(n−1)/nkfkp,
which, by (4.3), yields
k|∇u|knp/(n−p)≤c3k|F|knp/(n−p) ≤
c4
(p−1)(n−1)/nkfkp,
i.e., we get the assertion.
We are going to prove
Corollary 4.7. If α > 0, f ∈ L(logL)α, and let u be a solution of (4.1). Then the apriori estimate (4.2) holds for any α≥(n−1)/n.
Proof. It is clear that the mapping H which maps a given f ∈ Lp into |Du| ∈
Lnp/(n−p), where uis a solution of (4.1), is subadditive, satisfies the assumption
of Theorem 4.2 with r(p) = np/(n−p), p0 = n and β = (n−1)/n, by virtue
of Lemma 4.6. Hence Theorem 4.2 applies and, after extrapolation, using a standard density argument, we get the inequality for all f ∈L(logL)α, f ≥0.
Finally, writing f = f++f−, and calling u
+, u− a solutions of (4.1) with f
replaced by f+, f− respectively, we have
k|Du|kLn/(n−1)(logL)−1+[nα/(n−1)] =k|Du++Du−|kLn/(n−1)(logL)−1+[nα/(n−1)]
≤ k|Du+|kLn/(n−1)(logL)−1+[nα/(n−1)] +k|Du−|kLn/(n−1)(logL)−1+[nα/(n−1)]
≤Kkf+kL(logL)α +Kkf−kL(logL)α ≤2KkfkL(logL)α.
Acknowledgment
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(Received 17.09.2001)
Authors’ addresses:
A. Fiorenza
Dipartimento di Costruzioni e Metodi Matematici
in Architettura, Universit`a degli Studi di Napoli “Federico II” via Monteoliveto 3, 80134 Napoli
Italy
E-mail: [email protected]
M. Krbec
Institute of Mathematics
Academy of Sciences of the Czech Republic, ˇ
Zitn´a 25, 115 67 Prague 1 Czech Republic
